Direct Products of Closed Subsets

In this chapter we investigate various types of products arising naturally in scheme theory. We define direct products of closed subsets of S, direct products of schemes, quasi-direct products of schemes, and semidirect products of schemes. 

The relationship between quasi-direct products of schemes and direct products of closed subsets of S (as defined in the previous section) is described in the following theorem which generalizes [11 Proposition 3.13]. 

With the help of Theorem 7.1.3(i) we may refine both of the above representations of Ui to direct products of indecomposable closed subsets of S. Thus, as we are assuming that Ui T, we obtain, by induction, an element j in 1,..., n and an indecomposable closed subset W of Vj fl UAj such that... 

In the second section, we show that, if L is a finite set and does not contain thin elements, the closed subset generated by L is a direct product of simple closed subsets each of which is generated by the elements of L which it contains. 

The discussion above applies to uncontracted basis sets. Contracted basis sets present a few further problems. To properly represent the spin-orbit splitting, the two spin-orbit components should be contracted separately. The contraction is now j -dependent, rather than f-dependent, and can only be represented directly in a 2-spinor basis. The problem is not now confined to the small component. If the large-component scalar basis set includes contractions for both spin-orbit components, the product of the contracted basis functions for each spin-orbit component with the spin functions generates a representation for both spin-orbit components. Thus there is a duplication of the basis set that is close to linearly dependent, and some kind of scheme to project out linearly dependent components, either numerically or by conversion to a 2-spinor basis, is mandatory. The same applies to the small component. For example, the contracted p sets for the large-component and d sets both span the same space, but because of the contraction the (i-generated set cannot be made a subset of the -generated set, even if a dual family basis set is used. 

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